SHRP-C-627

Development of Transient PermeabilityTheory and Apparatus for Measurements

of Cementitious Materials

D.M. RoyB.E. Scheetz

J. PommersheimP.H. Licasttro

Materials Research LaboratoryThe Pennsylvania State University

University Park, Pennsylvania

Strategic Highway Research ProgramNational Research Council

Washington, DC 1993

SHRP-C-627Contract C-201

Program Manager: Don M. Harriott

Project Manager: Inam JawedProduction Editor: Marsha Barrett

Program Area Secretary: Ann Saccomano

April 1993

key words:

equipment designfluid flow

model developmentpermeability

pressure decaypressure differentialtransient pressure pulse

Strategic Highway Research ProgramNational Academy of Sciences2101 Constitution Avenue N.W.

Washington, DC 20418

(202) 334-3774

The publication of this report does not necessarily indicate approval or endorsement of the findings, opinions,conclusions, or recommendations either inferred or specifically expressed herein by the National Academy ofSciences, the United States Government, or the American Association of State Highway and TransportationOfficials or its member states.

© 1993 National Academy of Sciences

350/NAP/493

, Acknowledgments

The research described herein was supported by the Strategic Highway ResearchProgram (SHRP). SHRP is a unit of the National Research Council that was authorizedby section 128 of the Surface Transportation and Uniform Relocation Assistance Act of1987.

iii

4

, Contents

Acknowledgments ....................................... iii

Abstract ........................................... 1

Executive Summary ....................................... 3

Introduction ........................................... 5

Introduction to Theory of Pressure Pulse Testing ................... 5

Model Development ....................................... 6

Model Solution ......................................... 13

Discussion of Results of Mathematical Model .................... 14

Equipment Design ....................................... 16

Sampling Handling ...................................... 16

Typical Experimental Results ............................... 20

Limitation of the Equipment ................................ 20

References .......................................... 25

Appendix 1: Derivation of Equation (1) ....................... 27

• Appendix 2: Solution to Equation (8) ......................... 29

Abstract

A permeability apparatus was designed and constructed that would allow a rapid andaccurate measurement of water transport in concrete. The principle of the apparatuswas to subject a test specimen to a small pressure differential and to monitor both thepressure decay and the pressure rise resulting in response to the pressure pulse. Theapparatus consists of a design which can isolate two pressurized, large volume reservoirson both sides of the test specimen where the test specimen was acting as a permeablemembrane between the two. Very rapid measurements are possible with this apparatusfor specimens possessing permeabilities on the order of microdarcy to nanodarcy.However, in order to measure permeabilities below a nanodarcy the problems ofestablishing pressure equilibration throughout the test specimen becomes increasinglymore difficult.

To explain the behavior of cementitious materials subjected to pressure, mathematicalrelationships were developed building upon the work in existing literature on fluid flowin porous media subjected to a transient pressure pulse.

8D'ImUt&RY

The durability of concrete is frequently associated with the transport of dissolvedspecies. Such transport may be considered in terms of permeability. It is well recognizedthat transport occurs through a continuous network of pores, which exist in the

: cementitious matrix of concrete, as well as through the porosity which exists in theinterfacial regions of aggregate. It is the objective of this report to describe work leading torapid and accurate measurement of concrete permeability and the development of thetheory to describe permeability of concrete.

A transient pressure permeability apparatus was designed and constructed that wouldallow a rapid and accurate measurement of water transport in concrete. The apparatusconsists of a design which can be closed in a manner to isolate pressurized large volumereservoirs with the test specimen acting as a permeable membrane between the two. Adifferential pressure is established and both the pressure decay and the pressure rise aremonitored. Very rapid measurements are possible with this apparatus of specimenspossessing permeabflities of the order of microdarcy to nanodarcy. However, in order tomeasure permeabilities below a nanodarcy the problems of establishing pressureequilibration throughout the test specimen becomes increasingly more dlmcult.

In conjunction with the apparatus of construction, the detailed mathematicalrelationships building upon the original work of Brace et al. (1981), Lin (1977} and Hsiehet al. (1971) were also developed.

3

DEVEI_]PI_nrNT OF TRANSIENT pEp_r_.AmvY.rTyTHEORY ANDAPPARATUS FOR _an_._SU_.MI_-NT8 OF C_m_t-tTiOU-8 MATER/AI_

INTRODUCTION

-" The assllrnption that the permeability of concrete is a physical property measurementwhich can be used as an indicator of the quality of concrete has not changed since the outsetof this research program, Mortar samples prepared in this laboratory routinely achieve apermeability to water of <I0 nanodarcy which is equivalent to the values of 10"16 cm 2 forquality concrete such as those reported by Hooton (1988) and other workers.

The measurement of very low values of permeability presents special problems forwhich standard measuring techniques are g_neraIIy impractical or very difficult toimplement and therefore tend to be unreliable (Roy. 1988). If the permeability is very low,long periods of time are required to establish steady state flow conditions which are for themost part impractical. To overcome these limitations, Brace et al. (1968) introduced atransient flow method to measure permeability of Westerly granite to water. In thisexperimental design, cylindrical specimens of the granite were contained m a flexiblesleeve and connected to an up-stream and down-stream fluid reservoir. At the start of theexperiment, both reservoirs and the specimen were rn_Intained at the _rne constantpressure. Fluid flow was initiated through the specimen by rapidly establishing a pressuregradient between the up-stream and down-stream reservoirs. As the pressure began todecay through the sample, it was monitored and from this pressure decay, the permeabilitywas calculated.

Introduction to Theory ct Premure Pulse Testing

Transient decays in pressure have been used in the pressure-pulse permeability cell*to successfully measure hydraulic properties of rn_terials of low permeability such as coredsamples of rocks and sandstones (Brace et al., 1968; Hsleh et al., 1981; Ne1_l et al., 1981).To date, there has been little work devoted to the application of this technique tocementitious materials, which generally have higher compressibflities and permeabilities(Hooton and Wakeley, 1989). In the transient pressure pulse method described below, thejacketed sample is confined between two pressurized reservoirs which contain thepenetrating fluid. The pores of the sample are also filled with the fluid. The Jacket orconfining pressure is kept higher than the reservoir pressures. When the experimentbegins, the pressure in one of the reservoirs is suddenly changed to a higher or lower valueand the resultant pressure changes on the high or low pressure side are measured as afunction of time.

One of the best ways of collecting data is to simultaneously measure the pressurechange in both reservoirs. However, few researchers have used this method.Pommershelm and Scheetz (1989) have recently discussed the advantages of this technique.Hooton and Wakeley (1989) have emphasized the sensitivity of test results toenvironmental variables when measurement water permeabilities of concrete.

As part of a larger study made to characterize the microstructure and transportproperties of concrete, we have conducted experiments with cylindrical cores of cementsand mortars using both the pressure pulse cell and flow-through ceils in order to determineand compare their permeabflities. Further experimental details can be found in Part H ofthis report along with the application of the theory developed in this part to thedetermination of sample permeabflities from data collected in pressure pulse tests.

*Also known as the trlaxi_] or Hassler ceU.

5• °

Figure 1 presents a schematic diagram showing the experimental configuration forthe transient pressure pulse test. Before the test begins the entire system is at a constantpressure Po. Then the pressure in the upstre__m reservoir (Pu) is suddenly increased to P1,while the downstream reservoir (Pd) remains at Po. Pu will fall and Pd wlll rise as fluid istransferred between the reservoirs. A constant confining pressure (Pc) is kept outside thesample. By rnalntainmg this pressure at a level considerably higher than PI, leaks areprevented. However Pc cannot be kept too high because of the possibility of creep or mlcro-cracking or decreasing the permeability of the sample by closing otherwise open porosity.

Figure 2 is a schematic representation showing how the upstream and downstreampressures change with time for a material which typically has low permeability andcompressibilities. The figure illustrates the case where the two reservoir voliJmes, Vu andVd, are equal. Nomenclature is provided beneath the figure. The curves are the pressuredecay and pressure rise curves for the granite. Here the response of the two curves issymmetric around a horizontal line drawn to the final pressure Pf reached by bothreservoirs. As discussed by Pommersheim and Scheetz (1989), the analysis of Brace et al.(1968) predicts that this pressure will lie midway between P0 and PI, while the analysis ofHsieh et al. (1981) predicts that Pfwfll lle more towards the upstream side. The difference isattributable to the fact that Brace and coworkers assumed in their development that the

compressive storage of the s_mple was negllgible, whereas Hsieh and coworkers did not. Pfcan also be estimated from a mn.cs balance knowing the pore vol-me of the sample and thevolumes and initial pressures in the reservoirs (T_mmer, 1981).

Figure 3 illustrates typical transient pressure responses for cementitious materials,which usually have relatively high porosities, compressibilities, and permeabilities. Thepressure changes observed are more rapid for such materials and are not symmetric withone another. In this instance, it is observed that the final pressure is no longer equal to the

average of the initla] pressures but is skewed towards the hlgh pressure side of the pulsedreservoir. The rapid changes in pressure observed at early times appear to indicate that thesample is more compressible at higher pressures.

In order to explaln such apparently anomalous behavior it is necessary to firstdevelop and solve a mathematical model which predicts how the pressure changes withinthe sample and the reservoirs. The theory for this process is developed in this section andthen applied in the next, where general procedures are presented for determiningpermeabilities of cementitlous materials from experimental data.

Model Development

Previous mathematical models for pressure pulse testing have been developed byBrace et al. (1968), IAn (1977), Hsieh et al. (1981) and Pommersheim and Scheetz (1989).

As a model system consider the cylindrical sample depicted in Figure 1 presentedearlier, having total volume V = AL, where A is the cross-sectlonal area of the spe_men andL its length. The s_mple is confined between two pressurized reservoirs, the upstream oneat Pu and the downstr_m one at Pd. The initial values of the upstre__m and downstreampressures are PI and P0, respectively.

The partial differential equation which governs pressure changes as a function ofdistance and t_me P(x,t) within the .¢nmple is given by:.

_2p R,(_)2__u_ ,... _p (i)__-'_+v _x -k _t

6

SCHEMATIC DRAWING OF PERMEABILITYAPPARATUS DESIGN

P),:,,,, II

I , ,!! Ir"-ml

_,,=, Pd_---_ I I re6_o,rPc' P ' Pd

Figure 1. Schematic drawing of permeability apparatus.

7

%

Pu = up-strum pl"eSsul_Pd = down-stream pressurePf = final equilibrated pressure [-(PI + Po)/2]P1 = initial up-stream pressureP0 = initial down-stream pressure

Figure 2. Schematic representation of up-stream and down-stream pressure response as afunction of time during the experiment.

where P = pressure within the sample;_t = pore fluid viscosity;k = sample permeability;x = ax_l distance within the sample, measured from the upstream reservoir:,t = time;_' = lumped compressibility.

The derivation of Equation (1) is presented in Appendix 1. Equation (1) represents u

how the pressure P(x,t) varies within the sample as a function of position x and tlme t. It isa nonlinear partial differential equation subject to one tlme and two boundary conditions.These are given by:.

P(x,0} = Po initial condition

P(O,t) = Pu{t) boundary conditions

P{L,t) = Pd(t)

where L = sample thickness;Pu,Pd = upstream and downstream pressures, respectively.

Assumptions involved in the derivation of Equation (1) include: one dimensionalmass transfer with constant transport area, constant temperature, and constant physicalproperties (_, k, B' and porosity). These assllmptions are the same as those stated or impliedby previous workers (Brace et al., 1968, Hsieh et al., 1981). Most of them are likely to be metin laboratory tests with homogeneous corings. Sample compressibilities will be mostlikely to rern_In constant when the ratio of the confining pressure to the irdti_! pressuredifferent, i.e., Pc/(PI-P0) is high (Wooton and Wakeley, 1989).

_' is a 111mped compressibility. It depends on the compressibility of the fluid, B, thecompressibility of the -__mple, _s, the effective compressibility of the jacketed sample, Be,and the sample porosity, ¢, according to:

(2)

A similar equation has been presented by Brace et al. (1968) and by Hsieh et al. (1981). Of allthese quantities Be is the one which is least likely to be known and which is also potentiallythe largest, especially in experimental configurations where the sample is retalned in aflexible rubber or plastic sleeve. In effect this m_kes _le, and thus _', an arbitrary parameter.Neuzfl et al. (1981) found values for this parameter which were several orders of magnitudegreater than the fluid compressibility.

By introducing the dimensionless distance z = x/L, the O= t/T, and pressure P =(P-P0)/AP, Equation (1} a,nd its attendant conditions become:

(3)_z _ -_

(I) (II) (llI) °

where AP = P1 - P0,

10

• °

with conditions: p(0,O) = pu(0) reduced bounda_ conditions

p(1,8) = Pd(8) reduced initial condit/Qnp(z,0) = 0

T = _'ttL2/k is a characteristic time for the transfer of mass from the high pressure tothe low pressure side.

The terms in Equation (3) have been labeled (I), (II) and (I_). Term (1)represents theJ,

transfer of mass between reservoirs caused by the pressure difference. It is present in allformulations of the problem. The unsteady state term (III) corresponds to the acolmulationof mass within the sample. If the observed experimental tlmes t are much greater than thecharacteristic time T, i.e. t >> T, then the acc, lmulation term can be neglected. This is arequirement of a steady state process. However, since the pressures at the two ends of thesample are continuously changing, pressures within the system slowly adjust toaccommodate these changes, in effect adjusting to each new steady-state. Such behavior iscalled quasi-static and the system is said to be at quasi-steady state.

Term (II) in Equation (3) represents the dynamic response to the compression of thejacketed sample. This non-linear term will be small when the dimensionless

compressibility _'AP is small. Calculations show that term (If) would be small ff 9' is ofcomparable magnitude to the compressibility of the fluid, but, as discussed, this is often notthe case in practice.

Table 1 summarizes the mathematical solutions different workers have found inEquation (1) and its variants. In the solutions of Brace et al. (1968) and Pommersheim and

Scheetz (1989), only term (I)was retained. This predicts a linear pressure profile across thesample. The solution of Hsieh et al. (1981) is the only one which allows directly fortransient pressure changes within the sample. The others must rely on a quasi-steady stateanalysis in order to predict transient pressures within the reservoirs. However thetreatment of Hsieh et al. (1981) did not consider the non-llnear term (II). Their analyticalsolution was obtained by Laplace transform methods. It took the form of an infinite seriesand involved three dimensionless parameters: a dimensionless time and twodimensionless compressive storage or compressibility ratios.

To complete the formulation it is necessary to obtain conservation equations for theupstream and downstream reservoirs. These are shown as Equations (4) and (5),respectively:

dPu L _ V_( aP ix=0dt -T_f u _x [4)

d Pd L j_ _V2 Bx'_----_x=L (5)dt -T _fV d

where Vp = intruded pore volume;Vu = volume of upstream reservoir;Vd = volume of downstream reservoir:,

• _f = fluid compressibility.

These equations represent mass b_la_nces for the penetrating species. They are subjectto the initial conditions Pu(0) = P1 and Pd(0] = P0. Expressed in terms of almensionlessvariables. Equations (4) and (5) become Equations (6) and (7), respectively.

11

Table 1

Terms Present m Equation (3).

References Comment(I) (II) (II11

Brace et al. X quasi-steady(1968) •

Hsieh et al. X X ---(1981)

Pommersheim and Scheetz X quasi-steady(1989)

This work X X quasi-steady (Vu = Vd)

General solution X X X solution has not beenfound

12

dpu E_ Vp _z'_P_z--° (6)de - _f_Vu

dPd _' Vp _z'_P_z=l (7)" de -_Vu

where Pu and Pd are the d_rnens_onless pressur_ given by Pu = (Pu-PO)/&P and Pd =

(Pd-P0)/AP, respectively. The corresponding dlmensionless initial conditions on e are pu(0)= I and Pd(0)= 0.

The formulation of Equations (6) and (7) is slrnil_r to that of Hsieh et al. (1981). Inthis development the compressive storage of the reservoirs is taken equal to the fluidcompressibility since in practice the reservoirs are generally made from rigid thick-wagedmetal. This also insures that the boundary conditions do not contain a non-linear termcorresponding to the compressibility of the reservoirs themselves.

Equation (3) together with Equations (6) and (7) can be solved numerically to give pu(t)and Pd(t). This is the generaJ case indicated in Table 1. where a mlnlm,,m number ofsimplifying assumptions are made. The n, lmerical solution to this system of equations isnot trivial because Equation (3) is a non-llnear partial differentia/equation with timedependent boundary conditions.

As shown in Table 1 the solutlon presented in this work is a sub-case in which theaccumulation of mass [term (HI}]within the _mple is neglected but the non-linear term (If)is retained. As discussed, the accumulation term can be neglected ff observed experimental

times are much greater than the characteristic time, i.e., t >> T. In this sense T can beconsidered a system time constant, so that after a nl_mher of such periods have passed thesystem can be considered to have reached almost to steady state and ap/%0 _ 0 in Equation(3)."

Model Solution

Since the pressures at the two ends of the s_mple are continuously changing,pressures within the sample must slowly adjust to accommodate these changes. Underquasi-static conditions, Equation (3} reduces to:

d2p_2- + _'_P_= o (8)

with boundary conditions p{O}= Pu and p(1) = Pd.

Equations (4) and (5) are still applicable at quasi-steady state with the partialderivatives being replaced with ordinary derivatives.

An analytical solution to Equations (4). (5) and (6) can be obtained form the importantcase where the volume of the two reservoirs are equal, i.e., Vu = Yd. Det_il_ of the solutionare provided in Appendix 2. Equations (9), (10) and (i 1) present expressions for Pu, Pu-Pd

, and Pf, respectively:

° The validity of the quasi-steady state assumption and the application of this criterion tocementitious and other systerrL¢ is discussed more fully in Part II of this Report.

13

P1 " Pu 1 t/Tv

4_u- Ap -[_'AP) ] [1-ex_(-Y)]dy (9)o

where Y - 2 tanh- 1 [e-2Ytanh ( _ )

Pu "Pd Ylt/'l%']

AP - ([3'AP) [i0)

P1 -Pf 1

_f- Ap -{_'_p} [ [1-exp{-Y}ldy {111o

where v = V/Vp: with Vp = {intruded) pore volnme, V = Vu =Vd is the cou_ulon reservoirvolume and _ and ¢f are reduced pressures.

In implementing the solution, it is convenient to replace the term [I - expfY}] inEquations (9) and {11) by the equivalent form:

2n '2_1 -exp(-Y)= _'. (-11n+1 _-T{tanh-I [e-2Ytanh( ]}n (121n= 1

This series converges rapidly so that only a moderate number of terms of the summationare needed to guarantee convergence. At large values of the reduced compressibility theidentity (JoUey, 1961}

e2Y + tanh b/2

2 tanh "I [e'2y tanh b/2] = In [ e2Y. tanh b/2 ] (13}

is useful in evaluating the s,,mmAtion. To evaluate the integrals in Equations {9}and {1I), afour-point Guass-Quadrature integration scheme with variable step size and internal errordetection was used (Lewis}. In all cases the summation in Equation (12) was evaluatedbefore the integration was performed.

Discussion of Results of Mathen_tlc_ Model

Equations {9}through {13} can be used collectively to determine the predicted values ofPu and Pd at any time. It follows from Equation (10} that the upstream and downstreampressures approach one another at long times, reaching a common final pressure Pf. Pf canbe found using Equation (11). The final pressure is solely a function of the volume ratio vand the reduced compressibility {_'AP},while the pressures Pu and Pd also depend on thecharacteristic time.

Figure 4 presents a plot of the reduced final pressure ¢f vs. the reduced compressibility_'AP. At low values of _'tuP, _f-_ 1/2, indicating that Pf--_ {P0 + P1)/2. This agrecs withmodel predictions of other researchers {Brace et al., 1968, Hsieh et al., 198 i, Pommershetmand Scheetz, 1989}. As discussed in Appendix 1, this result can also be obtainedanalytically from a limit analysis on Equation (11). At high values of the reducedcompressibility, _ varies reciprocally with it according to the limit form:

_ [_l}n+l

{_'_d:')q_f---+] n_ 1 n! Inn{coth y} dy = kpo (14)0

14

Q0

00

i

°

Q

0 Q 0 "

0 0 0 0 @

t

4

N

15

Using the integral in Equation (12), the value of the transcendental number kpo to fivesignificant figures is 0.69314.

Figure 5 presents a plot of the reduced upst.r_m pressure ¢u as a function of thereduced tlme t/Tv, with the reduced compressibility _'AP shown as a parameter. Forsamples of higher compressibility the reduced pressure is less, indicating that the upstreampressure falls less rapidly for such samples. The final pressure reached (dashed asymptote)is greater than forsamples of lower compressibility, which agrees with the results illustrated in Figure 4. Theupper curve in Figure 5 shows the result for incompressible samples (_'AP = 0). This isidentical to the solution originally obtained by Brace et al. (1968) and later by otherworkers, for the case where Vu = Vd, that ks:

1 e_2t/,i_¢i_u1[3.___0= _ [1 - (15)

As discussed in Appendix 2, Equation (15) can be obtained analytically from Equation (9)using a limit ana/ysks.

Equipment Design

The permeability equipment including the cell was designed and initially constructedin the Materials Research Laboratory. Figure 1 presented earlier represents a schematicdrawing of the arrangement of the overall system showing the location of valves, the up-stream and down-stream reservoirs and the permeability cell. In this design, the pressurepulse ks applied to the system by rapidly reducing the pressure of the down-streamreservoir. Recovery time from this perturbation ks typically on the order of 10 to 15minutes.

Figure 6 ks an exploded diagram of the permeability s_mple cell showing the physicalarrangement of its parts. The current cell uses a thick 'Tygon" tube for the sleevingmaterial. This material has been found to be superior to rubber sleeving in that it does notreadily puncture under the influence of confining pressure in the presence of surfaceimperfections in the s_mple. Three cell sizes are available which can accommodatesamples of 1", 2" and 3" in diameter and lengths varying up to 6". Figure 7 ks a detailedengineering drawing of the assembled ceil.

The pressure response of the experlment is monitored electrically with SCHAEVITZpiezoelectric transducers designed to operate over a pressure range from 0 to i000 psi. Theemf output of the transducers is monitored on an IBM PC computer into which aMETRABYTE DAS-8 data acquisition and control board was installed.

The computer control of the data acquisition ks achieved with a compiled DOSalgorithm. The program reads the analog inputs to a file for storage which can beannotated with a descriptive text file for archiving purposes. Data acquisition time ksselected by the operator. The output file of this program ks stored separately on a 5.25"floppy disc and further processing ks accomplished by reading this raw data into anotherroutine called "convert" which transforms the raw data into a psi vs. time file which can beread into LOTUS 123 for final data processing.

Sample Handling

In order for this apparatus to work, all specimens to be tested must be fully watersaturated before the experiment is initiated. Water saturation ks achieved by vacuum

16

0n

C.. ,.,i

f

,0 i

_. o._ o I,l_ I I I I I IC_. eoQXOI

e,,-,l

!' +

.+u ,<, .= ,+.+,

_ m

?p mJ ,':

, _

-.. .+ , '.. _ .+-,°o o ow _ q. _ o _._

o d o o oc)

- _6

u_

17

[ TYGON SLEEVE

SAMPL(

O-RING

O-RINGS

Figure 6. Exploded diagram of permeability ceil.

18

I

Fastening rodRubber sleeve

)le

Confininq pressure

0 I 2I t •

cm$

To collection bottleLoad ceil - LVDTWeighing - Plotting

Figure 7. Sectional view of the liquid permeability apparatus.

19

impregnation; the _mple is placed in the vacuum produced by a roughing pump for 24hours before it is Immersed in delonized water. The water saturated specimen is thenplaced in the sleeving material and the cell assembled according to the detailed procedurewhich appears in the SHRP 201 Fourth Quarterly Report (1989).

Typical _perlmental Remxlts

Measurements have been made on several samples of the Tuscarora sandstone,actually an orthoquartzite, for calibration purposes. The initial theory adequatelydescribes the pressure decay characteristics and permeability. Figures 8a,b are examples ofthe output of the apparatus. Figure 8a represents the pressure readings for the upper- andlower-transducers recorded for the course of an experimental lasting approximately 1200minutes. When these data are processed as described above, a straight llne, as shown inFigure 8b, is produced.

Initial data collection for cement pastes and mortars has been inconsistent with themathematical theory upon which the experiment was designed. A considerable amount oftime has been expended in order to reconcile the differences between the observed"extended" exponential decay of these materials. The results of these experiments willrequire that additional modifications to the theory be made in order to adequatelyinterpret the observed pressure decay.

Measurements on several _mples of the quartzite have resulted in a permeability of9 x 10-8 darcys. This value is consistent with previous permeability measurementsmeasured using both liquid (water) and gas systems: 1.7 x 10 -7 and 3.6 x 10-7 darcysrespectively. The slightly lower value is attributed to differences in the confining pressuresbetween the two experiments. These comparisons strongly suggest that the apparatus isfunctioning properly. Further, the nature of the data collected for sandstone clearly show,at least for this kind of material, that the mathematicM analysis developed is valid.

A comparison of Panel A in Figures 8 and 9 reveal the same general form of thedecay/pressure rise transducers curves. The processed data for a cement paste are presentedin Figure 9, Panel B. The "extended" exponential form for the data differs from that of therock (Figure 8) and will require modification to the mathematical model.

T-lm_R.U_ Of the F4uipment

The apparatus has been designed to measure very low fluid flow, below the range of ananodarcy. This permeability region is of particular interest because many of our previoussamples of concrete and mortars had very low flow rates. As described earlier, theapparatus designed has been successful, however, we have seen that in order to obtainrepresentative data describing the flow of fluid through a concrete specimen, the pressuremust be equilibrated throughout the interior of the specimen. Equilibration of the pressureis monitored by establishing a reverse flow through the test s_mple. That is, Pu is placed ata lower pressure than Pd and then reversed before the test data is formally collected. Hereinlies the crux of the problem. The time required to reach this equilibrium is of the sameorder of magnitude as that required to conduct the experiment (Roy, 1988), which could beseveral seconds to minutes for specimens with flow greater tha_. a microdarcy or from daysto weeks or longer for sub-nanodarcy specimens. Figure I0 represents the times required tosaturate test specimens. The data in thls figure are characteristic only of one set ofconditions in which the sample is assumed to be one inch in diameter and one inch inlength with a totally dry porosity of 25%. The principal data were determined by solvingdarcy's law for the volume of water necessary to fill the 25% porosity at varyingpermeabilities and a fixed pressure differentla). As can be seen, times to saturation can be

2O

21

22

TIME TO SATURATIONLOG TIME (HRS)

1.0E+09 . :...........: .... _... .:......i ....

1.0E+08 _.iI ::,:!::.:.:..:._.!.:!.. : :.:..._.:_:. :.:.i ......:..:ii!i:!i_:.:.:.::.:.: -1.0E+07"!:_:/_._:_!!!_i::::_!:I_ ...._!_"=_n_,,__,:_a._.:_._.....'::__ .....'

1.0E*05 i ; :;:: iiil i .1.0E 4���....

_!i{!!!{:i:::!:::!::::-::::_::_::: -_::: =====================:

1.0E+02 . :t.1.0E+OI

1.0E+O0

1.0E-01........ : ..... :............ . ........ . ".':. : ".!: .1 ...'.. ._., ......: . . "_ . .

1.0E-02 _ ......... I.... : i .... I ....I; : ["""-14 -12 -10 -8 -6 -4 -2 0

LOG PERMEABILITY

- saturation time

BASED ON 25% POROSITY; COMPLETELY DRIED

Figure I0. Bounding calculation of the tlme necessary to saturate a completely dried testspecimen with 25% porosity. All calculations are based upon solving darcy'sequation at fixed permeabillties for the time necessary to fill the porosity at afixed pressure. The parameters used in this calculation are necessarily the sameas the test equipment design.

23

very long. The horizontal llne segment represents the practical llmlt of laboratory usagefrom this research. Attempts to enhance pressure eq-i1_bration by using elevatedhydrostatic pressure before conducting the data collection will only very slightly assist inreducing the time required to equilibrate the sample.

In an open porous system where only capillary action is responsible for backfll]Ingthe porosity, the following relation controls the fraction of the capillary volume Riled:

X = 1/[1 + (rP'/2Ycos_]]

where: X = fraction capillary volume filledr = critical pore radiusY = surface tension of water

E) = contact anglep' = pressure.

This relationship suggests that at atmospheric pressure only 91% of the volume of a singlecapillary 150 nm in radius will be filled with water under capillary action. If the pressureis raised to 700 psi, only 95% of the single capillary will be Riled. For interest in thispermeability applicaUon, the rate at which this capillary will fill is known as Poiseuille'sLaw:

L2 = Ncos0/2n]rt

where: L = penetration lengtht = timeY = surface tension of water

0 = contact angler = critical pore radiusn = viscosity of water.

To completely penetrate the single capillary 150 nm in radius, according to thlsrelationship, would take on the average 37 years.

Although these calculations represent only bounding approxlm_tion, they suggestthat very low values of permeability can only be achieved at the expense of lengthy samplepretreatment.

24

P_ccs

W.F. Brace, J.B. Walsh and.W.J. Frangos, '_Permeability of Granite Under High Pressure," J.Geophys. 1_s. 73 (6), 2225-2236 (1968).

J.D. Hooton and J.D. Wakeley, "Influence of Test Conditions on the Water Permeability ofConcrete in a Triaxial Cell," Materials Research Society Symposium P, 1988, MRS

Press, Pittsburgh, PA (m press).

P_. Hsieh, J.V. Tracy, C.E. Neuzfl, J.D. BredehoeR and S.E. Si||iman, "A TransientLaboratory Method for Determining the Hydraulic Properties of 'Tight' Rocks -- I.Theory," Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 18, 245-252 (1981).

L.B.W. Jolley, Summation of Series, 2nd Revised Ed.. Dover Publications, New York (1961).

W. Lin, "Compressible Fluid Flow Through Rocks of Variable Permeability," Report UCRL-52304, Lawrence Livermore Laboratory, University of C_llfornia, Livermore (1977).

C.E. Neuzfl, C. Cooley, S.E. SilIirnan, J.D. Bredehoeft and P.A. Hsieh, "A TransientLaboratory Method for Determining the Hydraulic Properties of 'Tight' Rocks m If.Application, Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 18, 253-258 (1981).

J. Pommersheim and B. Scheetz, "Extension of Standard Methods for MeasuringPermeabillties by Pressure Pulse Testing," to be submitted for publication (1989).

D.M. Roy, "Relationships Between Permeability, Porosity, Diffusion and Microstructure ofCement Pastes, Mortar and Concrete at Different Temperatures," Mater_l-¢ ResearchSociety Symposium P, 1988, MRS Press, Pittsburgh, PA (in press).

D. Trimmer, "Design Criteria for Laboratory Measurements of Low Permeability Rocks,"Geophys. Res. Lett. 8 (9), 973-975 (1981).

SHRP 201, Fourth Quarterly Report (1989).

25

APPENDIX 1: Derlvmtion of'Equation (1)

This appendix presents a derivation of the conservation equation for flow of acompressible liquid in a porous medium, expressed solely in terms of pressure.

The mass conservation or continuity equation for the liquid is:

_ =V. P v (A-11

where p is the liquid density and v is the superficial local fluid velocity, based on theoveraU cross-section. The constitutive rate equation for flow in porous med1_ is Darcy'sequation:

v =- VP (A-21

Substituting Equation (A-2) into (A-1):

_t - s]_ V- pVP (A-3)

Inthe above equations, the dependent variables 7, P and P are considered to be meanquantities, averaged over a section of flow large when compared to mean pore sizes.

From the phase rule p = f(T,p), it follows that:

dp= _fdP- _ (A-4)P

where _p, the volume expanslvity, is given by - _- p,

and _I', the isothermal compressibility, is P .

At constant temperature, Equation (A-5) can be integrated to give:

p= pp e_I "P (A-5)

where pp is the density at low pressure. In the dez_eation of Equation (A-6) it is presl,medthat the isothermal compressibility does not vary significantly with pressure over thepressure range employed.

Equation (A-5) can be considered to be the equation of state for the liquid, describinghow its density changes with pressure. Substituting it into Equation (A-3):

k_t =_ V_'p=_" _j_ (i-6)

Here the assumption if made of one dtmensionad flow (in the x direction) with constanttransport area.

The pressure form of Equation (A-6) can be obtained by substituting the equation ofstate (A-5). when:

27

_2p_+_T_2=_ _ (A-7)

Equation (A-7) is the same as Equation (I) of the text, with _rr taken equal to _', the s_mple ._compressibility.

28

A_n_xx _ so_tlcm of_ (s}

This appendix providesdetailstotheanalyticalsolutionofEquation (8)as givenbyEquations(9).(I0)and (II}.Equations(6}and (7}arealsoinvolvedsincetheyrepresentboundary conditions on Equation (8). The solution predicts reservoir pressures as afunction of time.

Making the transformation, q = dp/dz, Equation (8) becomes:

q --_ + _'APq2=0 (A-8}

At quasi-steadystate,thisequationissubjecttotheboundary conditionsp(O}= Pu and p(1)=Pd. Separatingvariablesand integratingtwicetorecoverp:

P-Pd_ In [c_8'AP+ z(l - e_'al_] {A-9)Pu - pd -_'AP

At low values of _'AP Equation (A-9} can be shown to reduce to:

P-Pd =l-z (A-10)Pu-Pd

Equation (A-10} predicts a linear pressure gradient across the sample. It is equivalent tothe result obtained by Brace et al. (1968}.

Application of Equations (6) and (7}, to Equation (A-9} yields Equation (A-12) and{A-13), respectively:

_ _Vu c_ I - exp [-_'AP(pu- Pd)] (A-1 i)_p dO-

dpd_p d8 -exp [-_'AP(Pu-Pd)]- 1 (A-12)

For the case when reservoir volllmes are equal (Vu = Vd = V) adding Equations (A-12) and

(A- 13} gives:

dn _'qp {exp [-(_'AP)u] - exp [(_'AP}u]} (A-13)dO -_fzV

u, the new dependent variable, is equal to Pu - Pd. Separating variables and integratingwith the initial conditions u(0) = 0, and expressing the result in terms of the originalvariables:

tanh [ (Pu- Pd)_' 2t2 - exp (-_-:_. tarth _ (A-14)¢

Equation (A-15) can be rearranged to given Equation (8) of the text. In the llmlt, as(_'AP) -_ 0, it can be shown using Equation (11) that Equation (A-15) reduces to the solutionfound by Pommershelm and Scheetz (1989):

29

In [ Pu "Pd 2tAp ] = "_"_ (A-15)

The other equations of the text can be derived in a s_rntl_r manner. Thus Equation (7)is obtained by substituting Equation (8) into Equation (A-12), separating variables andintegrating, while Equation (9) is simply the long t_me form of Eqation (7). Equation (13)results from Equation (7) by letting J3'AP-_ 0.

3O